Random Simplicial Complexes and Stein's Method

随着我们作为一个社会获得越来越多的数据，对数据分析的需求也在不断增长。有很多工具可以分析自然位于欧几里得空间的数据集。然而，并不是所有的数据集都采用这种形式。有大量的非欧几里得数据，比如说，以网络或流形的形式，我的目标是将统计学的研究与拓扑数据分析的研究结合起来研究这些数据。我对开发和分析可以应用于数据集的新方法很感兴趣，这些方法最好被认为是不会自然嵌入到欧几里得空间中的点的样本。对我来说，一个自然的起点似乎是网络研究，它在生物、社会科学、工程、化学、计算机科学、神经科学等领域有无数应用。一方面，有大量的概率和统计工具来分析现实生活中的网络和随机网络。另一方面，每个网络都是一个一维单纯复形，因此是一个拓扑空间，可以用拓扑数据分析的方法来研究它。此外，简单网络的空间本身可以被赋予度量，并且可以看作是一个拓扑空间。一个自然的问题是：假设有两个网络，我们如何比较它们？如果有一种算法，在给定两个网络的情况下，能够根据它们的内在结构对它们进行定量比较，那将非常有用。如果这样的算法对网络的结构做出最小的假设，甚至可以对大小和结构不同的网络起作用，那就更有用了。一种方法是在两个网络上定义过滤，应用持久同源算法，并为每个网络生成条形码。我们可以根据网络的拓扑摘要(即条形码)对网络进行比较。例如，条形码空间上有多个自然距离函数，如Wassestein或瓶颈距离，它们有稳定性定理等理论保证。这只是可用于比较网络的拓扑工具的一个示例。我的项目的第一步将是了解不同的网络拓扑比较如何在真实世界的数据集上进行经验性的工作，并使用统计和概率工具从理论上分析随机网络上的此类算法的输出。这将有助于网络分析领域以及随机图的分析。

The need for data analysis is ever-growing as we, as a society, acquire more and more data. There are plenty of tools to analyse datasets that naturally lie in an Euclidean space. However, far from all datasets take this form. There are plenty of non-Euclidean data, say, in the form of networks or manifolds, and it is my goal to combine research in statistics with the research in topological data analysis to study such data. I am interested in developing and analysing new methods that can be applied to datasets that are best thought of as samples of points that do not naturally embed into an Euclidean space.A natural starting point for me seems to be the study of networks, which has countless applications in the fields of biology, social sciences, engineering, chemistry, computer science, neuroscience, and many more. On one hand, there are plenty of probabilistic and statistical tools to analyse both real-life and random networks. On the other hand, each network is a one-dimensional simplicial complex and hence a topological space, which can be studied using techniques from topological data analysis. Moreover, the space of simple networks itself can be endowed with a metric and be viewed as a topological space. One natural question is: given two networks, how can we compare them? It would be very useful to have an algorithm that, given two networks, would be able to quantitatively compare them based on their intrinsic structure. It would be even more useful if such an algorithm made minimal assumptions about the structure of the networks, and would even work for networks that are different in size and structure.One way to go about it is to define a filtration on both of the networks, apply the persistent homology algorithm and produce barcodes for each of them. We could compare the networks based on their topological summaries (i.e. barcodes). For example, there are multiple natural distance functions on the space of barcodes like the Wassestein or the bottleneck distances, which come with theoretical guarantees like the stability theorem. This is just one example of a topological tool that can be used to compare networks. The first step in my project would be to see how different topological comparisons of networks work empirically on real-world datasets and also theoretically analyse outputs of such algorithms on random networks using statistical and probabilistic tools. This would contribute to the field of network analysis and well as the analysis of random graphs.